Which road shifters will work with Shimano Deore (M)610 Groupset?
Unfortunately, you can't use 10-speed Shimano road brifters with 10-speed Dyna-Sys mountain bike drivetrains. They just have different cable pull ratios.

You're limited to a few options if you want to run drop bars with 10-speed Shimano:


Mount your existing MTB trigger shift on your drop bars. This means you have to take your hands off the hoods and drops to shift. 
Get 10-speed Dyna-Sys compatible thumb shifters from Microshift. You must take your hands off the hoods and drops to shift.  
Get 10-speed Dyna-Sys compatible bar end shifters from Microshift. You must shift from the drops. 
Get Gevanalle GX brake/shifter kit, you must shift from the hoods using indexed bar end shifters mounted on front of brake levers.
You could try road brifters and a long cage road derailleur with a 10-speed mountain cassette. Definitely unsupported by Shimano. May be limited to 11–32 cassette instead of 11–34 or 11–36. YMMV.


However, you can mix and match road and mountain components with most SRAM 10-speed drivetrains. So you could use something like force shifters with X9 derailleurs with less trouble.