RSACryptoServiceProvider does work with SHA2-based signatures, but you have to invest some effort into it.
When you use a certificate to get your RSACryptoServiceProvider it really matters what's the underlying CryptoAPI provider. By default, when you create a certificate with 'makecert', it's "RSA-FULL" which only supports SHA1 hashes for signature. You need the new "RSA-AES" one that supports SHA2.
So, you can create your certificate with an additional option: -sp "Microsoft Enhanced RSA and AES Cryptographic Provider" (or an equivalent -sy 24) and then your code would look like (in .NET 4.0):
var rsa = signerCertificate.PrivateKey as RSACryptoServiceProvider;
byte[] signature = rsa.SignData(data, CryptoConfig.CreateFromName("SHA256"));
If you are unable to change the way your certificate is issued, there is a semi-ligitimate workaround that is based on the fact that by default RSACryptoServiceProvider is created with support for SHA2. So, the following code would also work, but it is a bit uglier: (what this code does is it creates a new RSACryptoServiceProvider and imports the keys from the one we got from the certificate)
public string Sign(string contentForSign,string priKey