Yes it definitely depends on the type and size of business that you are. For example, if you are selling advertising space, then a different person would be better. If you are in construction then it is okay for the account manager to follow up payment.
In terms of managing your cashflow, the most important thing is that you have a solid credit policy, and that you stick to it. If a payment is late by one day, then you should find out why ASAP.
If you were to go to an invoice finance company and ask them for a quote, you could see exactly how much each day a payment is late is worth. Large companies are aware of this and can take advantage of smaller businesses by dragging out payments (we see this everyday).