As Gary mentioned, so long as you set up your phone with the same account(s) you had on it before you flashed the ROM, the apps will download and install for you. Your apps follow your account, not your phone.
You can also use an app like Titanium Backup to backup your app(s) + settings and then restore them after flashing a new ROM. This is my preferred method, as you don't have to spend a lot of time on the settings again. The free version lets you do one backup total at any time, which is good enough for a casual user, and there is a batch mode to let you back them all up at once (I don't recommend doing that with system apps across ROMs though as you may get strange results). It does require root, but if you're flashing a new ROM anyway that shouldn't be an issue.