Cleanest way to write retry logic?

You should try Polly. It's a .NET library written by me that allows developers to express transient exception handling policies such as Retry, Retry Forever, Wait and Retry or Circuit Breaker in a fluent manner.

Example

Policy
    .Handle<SqlException>(ex => ex.Number == 1205)
    .Or<ArgumentException>(ex => ex.ParamName == "example")
    .WaitAndRetry(3, _ => TimeSpan.FromSeconds(3))
    .Execute(DoSomething);

Blanket catch statements that simply retry the same call can be dangerous if used as a general exception handling mechanism. Having said that, here's a lambda-based retry wrapper that you can use with any method. I chose to factor the number of retries and the retry timeout out as parameters for a bit more flexibility:

public static class Retry
{
    public static void Do(
        Action action,
        TimeSpan retryInterval,
        int maxAttemptCount = 3)
    {
        Do<object>(() =>
        {
            action();
            return null;
        }, retryInterval, maxAttemptCount);
    }

    public static T Do<T>(
        Func<T> action,
        TimeSpan retryInterval,
        int maxAttemptCount = 3)
    {
        var exceptions = new List<Exception>();

        for (int attempted = 0; attempted < maxAttemptCount; attempted++)
        {
            try
            {
                if (attempted > 0)
                {
                    Thread.Sleep(retryInterval);
                }
                return action();
            }
            catch (Exception ex)
            {
                exceptions.Add(ex);
            }
        }
        throw new AggregateException(exceptions);
    }
}

You can now use this utility method to perform retry logic:

Retry.Do(() => SomeFunctionThatCanFail(), TimeSpan.FromSeconds(1));

or:

Retry.Do(SomeFunctionThatCanFail, TimeSpan.FromSeconds(1));

or:

int result = Retry.Do(SomeFunctionWhichReturnsInt, TimeSpan.FromSeconds(1), 4);

Or you could even make an async overload.