I not a US citizen and don't have a Social Security Number but will be traveling to the US for three months, bringing cash.
Do any banks offer tourists a checking account with debit card, so I can deposit my cash and not have to pay in cash all the time?
I prefer checking accounts with the lowest fees obviously: lowest balance needed to be free, no transaction fees, low fees on depositing etc.
As far as getting a bank account, the simple answer is "it depends". Banks can legally open an account without an SSN (I've done it!), but many will not do so except under exceptional circumstances. However presuming you're spending the time you're in the US travelling around rather than being at a single address it's unlikely that most banks will allow you to open an account without some form of fixed address.
However there is a good alternative - a "reloadable" Pre-Paid Credit Card. These go under a number of different names such as Serve from American Express and GreenDot and many others who do cards from both Visa and Mastercard. These cards normally cost a few dollars to buy, and otherwise generally have no or very low fees (eg, Serve costs $1/month).
Note that these cards are NOT the same as the pre-paid "gift cards" many places sell. Those generally only come with a fixed amount of credit and can not be recharged, but more importantly if your card is lost or stolen, they can not be replaced like the cards mentioned above.
You don't mention where you are from, but a potentially better option is to look for cards in your home country that do not charge international fees. These used to be very uncommon, but in recent years more and more banks have started offering them.
No, you are not going to be able to get an account. U.S. banks are not very accommodating to non-citizens or citizens for that matter, especially since the “homeland” security acts. You must have a social security number and an I.D. with a U.S. address. You can buy “Visa debit cards” for a fixed amount that have replaced travelers checks, but there is an fee for using the cards that is about $7.00 each time the card is used.
I suggest that you get a debit card in your country, and set a daily limit on how much can be withdrawn, if you are worried about card theft. There are couple of things to consider, before traveling to the U.S. without a credit/debit card. One is you cannot rent a car in the U.S. with out a credit card. Second is that hotels/motels will sometimes charge you a huge deposit fee if you don't have a credit card. I'm an ex-pat and have run into these problems when I visited, Americans are friendly, but it's not the easiest place to travel.
