EDIT: I'm posting this in response to kaykurokawa since I don't have sufficient reputation to comment directly.
POST: This isn't correct. There is a lot of confusion about how Gridcoin's Proof-of-Research system works.
Gridcoin has two types of block awards - PoS, and PoR. PoS works as traditionally expected but it is used as a backup mechanism. The primary block reward mechanism is proof-of-research, and your chance to solve a PoR block and the amount of GRC you receive are dependent on your relative contribution to the BOINC network.
Note that the solution to the PoR block uses traditional hashing mechanisms (PoW) as far as I'm aware, but if your contribution to BOINC isn't high enough relative to the rest of the Gridcoin network participants, 1) you will not be awarded a block and 2) the Gridcoin client is smart enough not to waste CPU cycles trying and instead frees those cycles for BOINC (or whatever else you want to do with them).
If for some reason the BOINC network goes down or people totally stop researching, PoS is there merely to ensure that transactions will continue to be processed.
To more directly answer the original question, Gridcoin is built on a significantly modified Litecoin codebase, so from a programming perspective it's not exactly a cut-and-paste thing.
Theoretically, Bitcoin could be altered to use any other reward mechanism dreamed up and accepted by the community, but in practice Bitcoin is so established and people have made such significant investments into hardware and infrastructure that I think there would be insurmountable resistance to changing the way that block rewards are distributed.
In terms of consequences, the primary positive consequence is that the freed-up CPU cycles currently being used by Bitcoin for simple PoW hashing could be spent on more useful things like BOINC. On the negative side, there is some concern about Gridcoin's dependency on a second outside network (the various BOINC projects and statistics collections and so on) which could be an issue for people who believe that it negatively impacts Gridcoin's decentralization as compared to Bitcoin, which has no such external dependencies.
Note that Gridcoin seems to have plans in the works for allowing direct in-network distributed computing which could mediate that in the future, but I'm not sure how far along it is.
DISCLAIMER: I'm not a Gridcoin dev, just someone who really believes that it's a much better use of electricity than Bitcoin, much as I love Bitcoin.:)