Death to Spies?
The thematic epoch is obviously close, but it is none of the "Death to Spies" games.
Indiana Jones and the Emperor's Tomb?
Congratulation, it is indeed "Indiana Jones and the Emperor's Tomb
I played it a little bit in the past, but could not really get into it. Maybe, I liked the actual Tomb Raider formular more...
However, I gave it another chance, recently after finishing the Tomb Raider Legend, Underworld and Anniversary trilogy.
And I have to say, that I was very positively surprised not only by the brawling mechanics in the fist fights but even more so by the level design of various parts of the game--the second area (a castle in Prague) is a great example of a map where you get to explore different sections that are all interconnected; it allows the player to see either in advance in the distance or from a different vantage point (e.g. rooftops or a tower) areas you are visiting later or which you have passed through already. This gives a nice cohesive and authentic feel of each section of a level!
Unfortunately, this type of level design is scrapped in the later half of the game in favor of more throwaway "cinematic approach" of set pieces, which may be impressive individually, but are underutilized and more loosely connected with each other... (The villain's palace towards the end is an exception, though.)
Anyway... it is your turn now, Johnny_GT!