Can I use dish soap and warm or hot water to clean my kitchen drain

YES BUT ADD A DROP OF bLEACH TOO.

I would recommend using baking soda, then vinegar, wait ten to fifteen minutes, then wash down with a gallon of boiling water. This will take care of gunk in the pipes and clean the pipes. The vinegar has grease fighting properties as well as antibacterial and antifungal properties.

Since it works for nasty toilets, why not? Just make sure you don't go crazy with the soap.

baking soda and white vinegar works well - pour 1/2 c baking soda down the drain then follow with white vinegar, I usually heat mine up a little bit and it will clean out the drains

Grease and food debris tend to clog drains. If it is draining well, I would say yes. I keep a bottle of Dawn just for grease issues in my house, and it's an inexpensive method for dealing with grease.

Sure, why not? My dishwasher is out of commission at the moment, that's what I do at least 20 times a day.

Pour some baking soda in the drain and follow with some plain white vinegar.

Pour some boiling water, then 1/2c. baking soda and a couple cups white vinegar. After 15 mins., run HOT water for a minute. I do this weekly on all the drains...no clogs. P.S. I have a septic tank.

Baking soda and vinegar

http://www.hometalk.com/diy/clean/bathroom/diy-how-to-clean-your-drain-naturally-1235410

clean toilet with dish soap