Does Grover's algorithm effect block size or only key size? We know that Grover's algorithm can speed up cracking symmetric keys. Basically the keyspace is halved. This means we have to use at least a 256-bit key (to get 128-bit security).

I heard somewhere it also has an effect on the block size (so we should use 256-bit blocks instead of 128)!

Is that true?
It depends on the application. If you are using the block cipher as a hash function or for a MAC (say in CBC-MAC fashion), then it very well could create problems. Preimage attacks would be much easier than they should.

For normal encryption, however, there wouldn't be a problem since the key is not known to the attacker. As evidence of this, consider Triple DES which has a 64 bit block size yet is still sufficiently secure for use today (though I wouldn't recommend it in newer designs).