Just to be sure, it's your H visa which is expiring (after 3 years) not your LCA, right? The LCA (Labor Condition Application) is something which must be filed in order to obtain an H visa and (as was the case for me recently) if you move to a different location, the LCA needs to be redone (but the H visa stays the same). It's not the LCA which lets you work -- it's the H visa (or another visa) which was obtained for that LCA.
So, assuming it's your H visa which is expiring, and assuming you're not at the 6-year cap, your options should be to either renew it (which may or may not mean sending in the LCA again -- I'm not sure) or dropping the H and working under the EAD.
You can do pretty much anything with your EAD (including washing toilets at Wal*Mart in Alaska if you want). You shouldn't need an LCA as far as I know -- the EAD (Employment Authorization Document) pretty much allows you to work anywhere with no restrictions afaik (while the AOS is pending).
But, if you have the option of renewing your H visa and continue to work under it (this would really be up to your company / lawyer) you might want to do this.
The pros are:
+If your I-140 / I-485 is denied, you can continue to work under the H (otherwise you'd be out of status as your EAD would be invalid upon the denial).
+Won't need AP to travel (as you would with EAD). This was taking 6+ months to obtain last time I checked so if you want to travel before that, you should push to renew the H.
+More convenient -- should be good for another 3 years (and could be extended after that if necessary while the GC is pending). With the EAD, you have to renew this every 9-10 months or so to be sure that you're covered (and won't have to run around trying to get an iEAD). This applies to your company / lawyer too -- you won't need to hassle them about renewing the EAD, possibly 1-3x as you wait for the GC (hopefully the USCIS is trying to reduce the wait but at this point a 2-3 year wait still seems possible).
The cons are:
+You'll be using up your 6 year max on the H visa (should you want to revert to this at some point)
I currently have an EAD (no AP) and I'm working under an H visa.
ETA