A promise is a promise.
一言既出，驷马难追。

– Did you take that money?
– No,cross my heart!
— 你拿了钱吗?  
— 没有，我发誓!  

Cross my heart意为“我发誓”。

On my honor, I swear to youthat I did not steal that money!
以我的名誉担保， 我向你发誓我没有偷那笔钱!  

On (one’s) honor意为“以名誉担保”。

– But listen, you have to promise never to tell anyone.
–My word is my bond.
— 听着，你得保证不告诉任何人。 
—我说话算话。 

Bond在这里表示“契约，合约”，My word is my bond意为“我说话算数”。

Take my word for it – I will not let the information out.
相信我，我不会把消息泄露出去的。 

Your clients will distrust you if you keep promising them the earth. 
如果你总是给客户不切实际的承诺，客户就会不信任你。 

Promise (one) the earth意为“向(某人)作无法兑现的承诺”。