Absolutely keep logs and absolutely bring them to the surface and work them on a strict cadence. Especially risks. Without the logs, people will happily ignore them. There is a resistance in raising and working risks, I think because of the mantra that optimistic and can do people are more successful.
I have never maintained assumptions and dependencies logs because these are a source of risks. If an assumption is made, there is a risk that follows that and, keeping two logs, you would end up having the same exact event being tracked on two logs. Same for dependencies.
Issue and Risk logs have a very similar connection that is often missed. When you have an issue, you have downstream threats. When you log the issue, people tend to forget about the assessment of those downstream threats. And if you do the risk assessment, you could find yourself working the same business event in two logs. This is inefficient and opens the door to miscommunication.
Edit: I have found in facilitating risk working sessions that we end up in this ongoing argument as to whether an identified event is either a risk or issue. It comes down to how an individual was framing the event, either tactically or strategically.
For example, a patient presents to his doctor with high blood pressure and cholesterol, over weight, family history of heart disease, and sedentary life style. All of these are issues based on the normal definition of an issue. However, all of these are also risk factors for the risk of heart attack. The heart attack is not certain nor its severity. If this was a project, would you log the BP, cholesterol, obesity, family hx, lack of exercise in the issue log or the heart attack in the risk log...or both?
Tactically, you would treat the issues; strategically, you would mitigate the heart attack as well as plan for after the heart attack by way of contingency planning. Treating the issues and mitigating the heart attack are likely the exact same actions but, the way we approach it in projects, we have a tendency to over simplify this and attempt to treat these things separately, causing disagreement and argument and inefficiencies. We tend to lose sight of the bigger picture, I have found.
I think trying to separate risks and issues is a mistake and maybe we should approach this more as exceptional events. Sometimes we are tactical, sometimes strategic, sometimes both.