Transferring to a new host

Due to various problems I’m having with my WordPress install (host issues rather than WordPress itself), I was wondering what’s the cleanest way I can move my WordPress install over to a new host?

What I’m thinking is whether I can download WordPress and reinstall from scratch, copy over my theme and then transfer the MySQL tables. Obviously, I’ll need to reinstall my plugins and play with the various settings, but will this work? Will transferring the MySQL tables over restore my posts, comments, etc, on a different WordPress install?

Thanks,
David.