Web of Science turns up some 143 papers on 'cicada emergence' across all search fields. There are a dozen or so that focus on the impact of the emergence on various predators, and a variety that focus more on the impact of the emergent biomass on the environment.
Lets start with a review article from 1995 by Williams and Simon. Here they state
Possibly the most important ecosystem function of periodical cicada emergence
may be providing increased energy and nutrients to predators. ... In periodical cicada emergence years, avian predators
showed increased fledgling success (4, 34, 1 16), nestling biomass ( 1 16), and clutch numbers and sizes (4, 34, 85). ... Krohne et al (60) found that periodical
cicadas enhanced reproduction of some mammalian predators (e.g. shrews,
Blarina spp.), but not others (e.g. mice, Peromyscus spp.). Hahus & Smith
(29) also found that Blarina, Peromyscus, and Microtus (voles) incorporated
periodical cicadas into their diet, but cicada abundance seemed to produce
increased intake of arthropods only for Peromyscus.
Certainly lots of things eat cicadas, and their primary predation avoidance method is to 'satiate' the predators - too many cicadas, not enough eaters. The review article above notes that
consumed only an estimated 15-40% of the cicada population, and much of
that probably occurred after cicadas had reproduced.
A study focused on the impact on rodents states
Data from a non-emergence year revealed no differences other than the presence of periodical cicadas that could explain the 50% higher relative densities of P. leucopus in the emergence year.
Or, white-footed mice love to eat cicadas and their population booms that year. Nothing is said about long term dynamics.
An article on bats by Storm and Whitaker notes that
Although periodical cicadas constituted a fair amount of the diet during the emergence, the dietary diversity of big brown bats during the emergence was similar to that of non-cicada years.
but makes no statements on the impact on population because of the availability of the cicadas. They don't specifically hunt them down, but will eat them when they find one.
An article by Koenig and Liebhold makes the statement that
Here we show that periodical cicada emergences appear to set populations of potential avian predators on numerical trajectories that result in significantly lower potential predation pressure during the subsequent emergence.
The same authors note in a more detailed paper on avian populations that
These results suggest that the pulses of resources available at 13- or 17-year intervals when periodical cicadas emerge have significant demographic effects on key avian predators, mostly during or immediately after emergences, but in some cases apparently years following emergence events.
with some species showing lower population numbers in the years before emergences (wood thrushes and mockingbirds), while cardinals and house sparrows showed higher populations.
Summary: lots of things eat cicadas, but can't eat them all. Populations of various predators do change as a result of all that food. That last article is perhaps the most interesting, where long-term effects (on the order of the cicada cycle) are noted on various bird species.