There is nothing in the "Snowden revelations" which even hints at any special NSA ability at breaking RSA. Even taking all that Snowden says as gospel, NSA is still at the same point as everybody else, meaning that breaking 1024-bit RSA is "theoretically feasible" but subject to the building of a very special machine whose design has been roughly sketched, which would cost a substantial number of millions of dollars, and, crucially, which would need quite some time (months...) to break every single key. For a 2048-bit RSA key, just forget it.
According to all the Snowden-powered information, when NSA "breaks SSL", they do it sensibly, i.e. not by punching through the cryptography upfront. Instead, they bribe the hosting sites to put spying hooks directly on the clear data, SSL notwithstanding. This is way cheaper, works reliably, and does not require invoking sci-fi level supposed cryptanalytic advances or alien technology.
Building and using RSA-breaking technology for mass-SSL spying would strike me as an extremely inefficient use of budget by the NSA.
Nevertheless, there are alternatives to RSA, and some have been part of SSL since the days of SSL 3.0, two decades ago. With the standard core protocol (called TLS 1.2 these days), you have the DH_DSS and DHE_DSS cipher suites, which involve Diffie-Hellman for key exchange, and DSA for signatures (with DH_DSS, the server's certificate contains a DH public key and the issuing CA contains a DSA public key; with DHE_DSS the server's certificate contains a DSA public key and the DH key pair is produced on-the-fly). Elliptic-curve variants have also been defined. Modern browsers and Web servers support DHE_DSS and elliptic curve variants thereof.
Historically, these alternatives were developed not for security, to fix some perceived or alleged weakness of RSA; they were put into SSL because at that time, RSA was still patented in the USA, and none other than the US federal government needed a patent-free but secure protocol for its own usages.
Now that the RSA patent has expired (more than 12 years ago), everybody does RSA. Yet alternatives are ready, and will be used if RSA appears broken. Which it currently does not, even against NSA.