EMDR is one of only two systems widely endorsed as highly effective, first-line therapies for PTSD, the other being TFCBT (prolonged exposure). As such, EMDR is superior to all other researched systems than TFCBT, to which it is roughly equal.
That said, EMDR is demonstrably gentler than TFCBT, requiring much less painful exposure to distressing traumatic material. It is therefore less prone to dropout. EMDR is also probably faster than TFCBT. There is some research and widespread anecdotal evidence for EMDR's relatively low drop-out rate and speed.
As for eye movements, the latest meta-studies and reviews judge EMDR's eye movements to be essential to its effectiveness and theoretically justified. There is much more to EMDR than eye movements, however, and the eye movement component arguably acts as a catalyst that makes the entire system faster and more effective.
First, in answering questions about EMDR, it's best not to rely on meta-studies and reviews prior to about 2003, since a lot happened after that to change the scientific view of EMDR, all in the positive direction. Even an important 2007-8 report is now out of date (see below).
Next, you are correct to say that eye movements or other bilateral stimulation (BLS) techniques are an aspect of EMDR. There is indeed much more to EMDR than BLS. (See Hyer & Brandsma, 1997; Rozelle & Lewis, 2014). But for that reason you cannot compare eye movements to other therapies -- that is compare a technique that is a part of a therapy to other complete therapies -- it is an ill-formed question.
Therefore, I assume you are really asking one or both of two related questions: (a) Is EMDR as effective (or more effective) than other therapies? (b) how important to EMDR's effect is BLS? Both these questions are addressed in the scientific literature.
The answer to (a) is well-established by numerous studies and judgments of professional bodies such as the International Society for Traumatic Stress Studies (ISTSS, the major international professional society concerned with PTSD) and national bodies in many countries (for example, the US Veterans Administration) -- EMDR is one of two highly recommended, first-line therapies for PTSD, the other being what's called trauma-focused (or exposure-based) cognitive-behavioral therapy (TFCBT) of which there are several forms. And between EMDR and TFCBT, neither is preferred based on the evidence. See Spates, etal (2010), pp. 279-301 for a summary of the evidence on EMDR published by the ISTSS, bearing in mind that the section on the efficacy of eye movements is out of date (see below). But on EMDR as a whole, this ISTSS report says "Based an on this review of seven new controlled efficacy/effectiveness studies of EMDR and seven new meta-analytic investigations of this technique, we assigned EMDR for treatment of adult PTSD and AHCPR rating of Level A." and that it is "equally effective as exposure-based therapies" (p. 298). In fact, "recent investigations... suggest the possibility that the dosed exposure, along with postexposure 'mindfulness' features comprising EMDR, might confer advantages over conventional prolonged exposure to trauma memories. However, this hypothesis requires substantially more research to test its validity." (p. 300)
As an example, Nijdam, etal (2012), a recent rigorous study with high n (140), concluded "EMDR and brief eclectic psychotherapy [BEPP, a form of TFCBT] had equal effects in terms of reduction of self-reported and clinician-rated PTSD symptoms, depressive symptoms and general anxiety symptoms when statistically controlling for pre-treatment differences. Across all outcomes, the response pattern was significantly different for EMDR and brief eclectic psychotherapy when accounting for baseline differences, indicating that EMDR led to faster symptom decline and brief eclectic psychotherapy to more gradual improvement." Not only was this an exceptionally rigorous and large study with controls against bias, it was conducted by the group at the University of Amsterdam that developed BEPP. So if there is any bias it would have gone against EMDR! This study is openly available at http://bjp.rcpsych.org/content/200/3/224.long. By the way, in this study EMDR worked approximately twice as fast as BEPP.
The only significant dissenting body is the US Institute of Medicine (IOM). IOM's 2007-8 report did not cast any explicit doubt on EMDR's efficacy, but simply excluded it from the top rank because it did not have enough studies that met the topmost criterion for rigor. But that extremely high bar for studies did not bother the other bodies, particularly ISTSS. Also, there are quite a few more studies on EMDR since the IOM's report, all positive. For a detailed discussion of the controversies surrounding all the institutional assessments of EMDR, see Forbes, etal (2010), pp. 550-551.
As for (b), the latest meta-study on the contribution of eye movements to EMDR's effect is Lee & Cuijpers (2012). They found (p. 231) "The effect size for the additive effect of eye movements in EMDR treatment studies was moderate and significant (Cohen’s d = 0.41). For the ... laboratory studies the effect size was large and significant (d = 0.74)." They also found methodological flaws in two major earlier meta-studies on eye movements, Davidson & Parker (2001) and Devilly (2002), and updated them with the considerable subsequent research.
Jeffries & Davis (2013) reviewed the literature on eye movements in EMDR and concluded "the results suggest support for the contention that EMs are essential to this therapy and that a theoretical rationale exists for their use. Choice of EMDR over trauma-focused CBT should therefore remain a matter of patient choice and clinician expertise; it is suggested, however, that EMs may be more effective at reducing distress, and thereby allow other components of treatment to take place."
One interpretation (in Rozelle & Lewis, 2014 forthcoming) of the recent results on eye movements in EMDR is that BLS is actually a kind of catalyst that enhances the actual core process of EMDR, that being information processing via mostly silent free association (Solomon & Shapiro, 2008, p. 321). Thus EMDR can indeed work without BLS, but it works better, often much better with BLS. In practice, however, there is no reason not to use BLS when doing EMDR -- it does help, sometimes a lot, and has not, to my knowledge, been shown to have any problematic side-effects.
Some last thoughts. EMDR is still held in low esteem by some academics, mainly because its early claims for efficacy seemed excessive and early research was not very rigorous. Both of these are now put to rest, but academics, like everybody, hate to admit their mistakes in public, especially when their earlier assertions were so strongly worded, and some even vituperative. Also, in those days waving fingers in front of a client's face seemed, on the face of it (pun intended) to be just too weird to be respectable -- not a terribly scientific opinion, but it carried a lot of emotional weight. Now that various forms of meditation have gained significant currency in psychotherapy, however, especially Kabat-Zinn's Buddhist-inspired mindfulness techniques, perhaps it's time to rethink the weirdness objection to BLS. How much weirder, after all, is attending to a therapist's fingers going back and forth than attending to your own breath going in and out?! (In fact, there may well be a relationship between those two techniques.) To tell the truth, nobody really yet understands in a scientific reductionist sense, why or how either works (not to mention any psychotherapy system)... but they do!
Davidson, P. R., & Parker, K. . C. (2001). Eye Movement Desensitization and Reprocessing (EMDR): A meta-analysis. Journal of Consulting and Clinical Psychology, 69(2), 305–316.
Devilly, G. J. (2002). Eye movement desensitization and reprocessing: a chronology
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Hyer, L., & Brandsma, J. M. (1997). EMDR minus eye movements equals good psychotherapy. Journal of Traumatic Stress, 10(3), 515–22.
Fiona W. Jeffries, Fiona W. & Davis, Paul (2013). What is the Role of Eye Movements in Eye Movement Desensitization and Reprocessing (EMDR) for Post-Traumatic Stress Disorder (PTSD)? A Review. Behavioural and Cognitive Psychotherapy, 41, pp 290-300. doi:10.1017/S1352465812000793.
Lee, C. W., & Cuijpers, P. (2012). A meta-analysis of the contribution of eye movements in processing emotional memories. Journal of Behavior Therapy and Experimental Psychiatry, 44(2), 231–239. doi:10.1016/j.jbtep.2012.11.001
Nijdam MJ, Gersons BP, Reitsma JB, de Jongh A, Olff (2012). Brief eclectic psychotherapy v. eye movement desensitisation and reprocessing therapy for post-traumatic stress disorder: randomised controlled trial. British Journal of Psychiatry, 2012 Mar;200(3):224-31. doi: 10.1192/bjp.bp.111.099234. Epub 2012 Feb 9.
Rozelle, D. & Lewis, D (forthcoming in 2014). What can Buddhism and EMDR tell us about treating PTSD? In V. Follette, J. Briere, D. Rozelle, J. Hopper, & D. Rome (Eds.), Contemplative practices in trauma treatment: Mindfulness and other approaches. Guilford Press, forthcoming 2014.
Solomon, R. M., & Shapiro, F. (2008). EMDR and the adaptive information processing model: Potential mechanisms of change. Journal of EMDR Practice and Research, 2(4), 315–325. doi:10.1891/1933-318.104.22.1685
Spates, C. R., Koch, E., Cusack, K., Pagoto, S., & Waller, S. (2010). Eye movement desentization and reprocessing. In E. B. Foa, M. Keane, Terence, M. J. Friedman, & J. A. Cohen (Eds.), Effective treatments for PTSD: Practice guidelines from the International Society for Traumatic Stress Studies (pp. 279–305). Guilford Press.