Runtime Verification (RV) techniques are becoming increasingly popular, due to their scalability and ability to analyse block-box systems. Monitoring is at the heart of RV; a logical formula φ, formalising some property of interest, is typically translated into a monitor that checks whether the system under scrutiny satisfies φ during its execution. A logical formula φ is violation (resp. satisfaction) monitorable iff there exists a monitor for φ that is both sound and complete w.r.t. its violation (resp. satisfaction) by a system; and φ is monitorable if it is violation or satisfaction monitorable. The monitorability problem naturally arises: given a logic L, what is the largest subset of L that is monitorable? While this problem has been solved for expressive untimed logics, it remains open for timed logics, i.e., where formulae can express both the order of events and the quantity of time separating them. In this paper, we solve the monitorability problem for Tlin, a new expressive (linear-time) timed μ-calculus that we propose. First, we show that Tlin is strictly more expressive than MTL, a popular timed extension of the well-known LTL. Second, we identify MTlin, the largest monitorable fragment of Tlin; and further characterise its largest subsets of formulae that are violation monitorable, satisfaction monitorable, and complete monitorable (both satisfaction and violation monitorable). To the best of our knowledge, this is the first work that answers the monitorability question for an expressive timed logic. Our theoretical results are accompanied with compilers (i) from MTL to Tlin and (ii) from MTlin formulae to monitors.