In any Debian-based distro (including Ubuntu, Mint, etc.), I like to use Synaptic for this. I go to Synaptic and do a search using the search term linux-image, then right-click on the old kernel (the name will start with "linux-image...") and click on "Mark for Complete Removal." Then I do a search using linux-header and mark the old header for complete removal. Then I click the "Apply" button. That gets rid of the old kernel and it also updates grub.
I try to leave at least one old kernel there, and of course the latest one as well.
Sorry, but if you don't use Synaptic in LMDE, somebody else will have to post some alternative instructions, but this is how I always do it.