Grab some of the disinfecting wipes in the round containers that you normally use for kitchen counters, etc.
I like the Clorox brand in the yellow
(Lemon Fresh!).
Wrap around bars and squeeze a bit as you twist/rotate in the same direction as the wrap.
With cork based wrap this is fairly effective.
With some of the slicker surfaced wraps (Lizard, etc), it's downright awesome.
If you start cleaning with the wipes on a regular schedule your wrap will get even cleaner over time.
bicycles.stackexchange.com/users/1036/Ken Hiatt
I use a citrus degreaser and its great.
I clean my entire bike with it, but it gets the sweat and oils off the tape
so the dirt stops sticking.
bicycles.stackexchange.com/users/3124/Matt Adams
I use a cleaning product called Simple Green.
It's bio-degradable ( good for the environment ) and really does a complete cleaning job!
You can mix it at different strengths from a 1/2 gallon or 1 gallon refill size.
It comes in a spray bottle initially.
I use it to clean my bicycle rather than soap and water.
It is safe to use and doesn't harm the paint.
Just spray it on the bike or a soft cloth, and wipe the grease and dirt/dust off easily.
No need to rinse the bike.
It should gently remove the dirt/sweat from your handlebar tape.
If the grips are cork, use just a little Simple Green on a soft cloth and wipe gently and remove from the tape.
Don't allow the Simple Green to stay on the tape for an extended period ( which is determined by the strength you mix the Simple Green ).
Same goes for wiping the bike using Simple Green.
Don't allow it to dry on the paint.
Wipe it on and with a dampened second cloth, wipe it off.
You'll find more uses for Simple Green besides your bike!
If you're married, better buy two bottles so your spouse can have their own bottle!
bicycles.stackexchange.com/users/None/
Dishwashing liquid and a small amount of baking soda, warm water, on an old sock or rag.
Wipe with damp rag.
Clean and bright.
Most of the dirt on my handlebar tape is from cleaning debris from tires while cycling.
bicycles.stackexchange.com/users/25538/user25538
I've used Cif or Jif as it used to be called.
Its a cream bathroom and kitchen surface cleaner.
Works wonders.
bicycles.stackexchange.com/users/16979/OraNob
I've never gone wrong cleaning wraps using a soapy rag and elbow grease.
I use an old bit of towel and dish soap.
This has done a great job cleaning up thoroughly bike-greased and sweat-upon wraps.
bicycles.stackexchange.com/users/2824/Darth Egregious
baby wipes all day son - best, cheapest option to rim to rim, and you can even get them in organic!
I've even used coconut oil as an on the spot bike lube.
bicycles.stackexchange.com/users/21052/face
I have bought not so long ago (about two month) brand new and the shiny white from my handle tap have become pretty dirty:    I am tempted to just change the tape to another color - that would be problem solved for me - but that would basically be a dirty hack. How can I clean the white tape and remove the dirt?
How to clean white handlebar tape?
bicycles.stackexchange.com
<cleaning><handlebar-tape>
bicycles.stackexchange.com/questions/11216
bicycles.stackexchange.com/users/1067/Chmouel Boudjnah
Various products are recommended for cleaning handlebar wrap. These include disinfecting wipes, baby wipes, citrus degreaser, Simple Green, dishwashing liquid with baking soda and Cif. Regular cleaning with cleaning wipes is an effective solution. A damp cloth and a cleaning product like citrus degreaser, Simple Green or dishwashing liquid and baking soda can also work.
Regular cleaning with cleaning wipes is an effective solution. A damp cloth and a cleaning product like citrus degreaser, Simple Green or dishwashing liquid and baking soda can also work.