I'd like to move untaxed investment money to a new institution. Originally, the money was held in a 401k, then moved to a Rollover IRA.
My new bank has both "Traditional IRA" and "Roth IRA" products and they say they're happy to receive the money but won't talk to me about the tax implications.
Are either of these types of account appropriate places to move a Rollover IRA, or do I explicitly need another Rollover IRA account to avoid this money from being converted into income?
You will need to move it to a Traditional IRA if you want to avoid incurring any taxes in the current year. A Roth IRA is for after tax money.
Also, you need to be very careful about how you do the transfer, there are very specific rules about how to transfer the money so it doesn't count as an early distribution and incur taxes and penalties. It is best to let the institutions handle the transfers directly rather than having the money come through you. This is called a trustee-to-trustee transfer
If they sent you a check already you have a very short window (60 days) to get the money in a new IRA equivalent account or incur fees and taxes. Ideally they should never write a check directly to you to avoid problems.
Yes, you can move it to a Traditional IRA per JohnFx's answer without incurring penalties.
However, one big thing to note is that if a future employer offers a 401k and you wish to move your money into it then they may be reluctant to accept money from a Traditional IRA into a 401k. Whereas moving from a Rollover IRA to a 401k is generally allowed with no issues.
Per https://finance.zacks.com/rollover-ira-vs-traditional-ira-2127.html
Choosing to create and maintain a separate rollover IRA account keeps open your option of a reverse rollover — a transfer of your rolled-over assets from the rollover IRA into a new employer's retirement plan.
