Can you have a sentence like
________________, I would often tell you.
and the part that's left out isn't in quotes?
I'm fairly certain the answer is, 'no'.
To use the verb 'tell' without quotation marks, you require an infinitive or a conjunction such as 'that'. This creates reported/indirect speech. Here are some examples:
He would often tell me that he was going to die.
He would often tell me to be careful crossing the road.
If you were to rewrite these using direct speech, they would become:
"I am going to die," he would often tell me.
"Be careful crossing the road," he would often tell me.
To remove the quotation marks then, you would have to begin the sentences with the infinitive or the adverb, which is very uncomfortable:
That he was going to die, he would often tell me.
To be careful crossing the road, he would often tell me.
I would say that neither of these are recommended constructions.
We may begin sentences with these clauses, but only if they are treated as nouns:
That he was going to die was very clear to all who studied him - where the first phrase is a fact or a piece of knowledge, not a quote.
To be careful crossing the road is something he would always advise. - where the first phrase is the something, an action treated as a noun.
I hope that helps.
Ok, this is a stretch, but if you allow for sardonic usage: