My wife has been mulling having me pull up our vinyl floor in the kitchen and putting down a hardwood floor. Hardwood is 3/4" thick, if memory serves and my toe kick is currently 4". So after I install the hardwood my toe kicks will only be 3 1/4". Is that a sufficient height or will I need to remove my cabinets and apply the hardwood under them?

That is up to you.

Standard Toe kick height, I believe is 4 1/4", or there abouts.

My kitchen's toe kicks are 3 1/2" with no added flooring on it (1950's Builder-built/lived-in house)

If you can live with 3 1/4"...that is up to you...

(Be aware that your dish washer may be affected by the hardwood flooring essentially 'locking' it under the counter - something to also take into consideration)

 

...(Be aware that your dish washer may be affected by the hardwood flooring essentially 'locking' it under the counter - something to also take into consideration)

Don't take this lightly. SIL bought a new-to-her house, and whouldn't you know a month or so after she bought the house, the DW shot craps. How'd we get it out? removed the counter top...what a PITA!

Rip