Modern-day equivalent of "dog my cats"

"Dog my cats" is a minced oath, reflecting oaths of the type "God damn my eyes", or "Damn my soul". It was moderately common in the 19th century (you can generally trust Twain); just Google Books the phrase for that period.

It's not entirely obsolete, but in contemporary speech it's used pretty much jocularly, with quotes around it, like "Well hesh my mouf" or "Jeez Louise" or "Holy Maloney".

We don't worry so much about vulgar language as Twain's readers did, so contemporary equivalents would be less reticent and less colorful. If you need something generic, "Heavens" or "For heavens' sake" have been pretty much standard since Jacobean theatres and publishers started using it to replace the terms forbidden by the 1606 Act to Restrain Abuses of Players.

My great-grandfather, born 1865, used to use the term and I asked him what it meant. He told me that it was short for 'turn your dogs loose on my cats why don't you'. It was used when someone was being unnecessarily mean to you verbally or physically.