Many reputable casinos do offer introductory NDBs, but most ask for a verification deposit in order to pay your winnings. This has a couple of reasons .... to prevent fraud and to establish correct payment information. Any of the CM accredited casinos that offer a NDB will not "scam" you when you make that deposit to collect your winnings. If you wish to withdraw what you won on a free bonus, make sure you don't take a bonus on your verification deposit. Then you can withdraw it and your winnings from the NDB. Make sure you read the terms and play according to them when you take any bonus, or you can find winnings voided.
While you may not have to deposit to try a NDB, they do have terms like you must play X number of bets, only play slots, bet no more than $5 a spin or maybe 10% of your bankroll, etc. An important one that is out there is only one SUB across all casinos in a group sometimes. Terms will vary from casino, so it's important to read each of them carefully.
While 3Dice does not offer a NDB, they do offer free tournaments where you can win real money, subject to some playthrough requirements. Unlike NDB offers, you are not limited to only trying once.
If you are going to try playing free offers, I suggest you open up an ewallet. Skrill and Neteller are a couple of popular choices. This way, you can make a withdrawal (and a verification deposit if necessary) easily.
You will also be expect to provide government issued photo ID and a recent utility bill or bank statement as part of the verification process casinos use to prevent fraud.