I'm not necessarily talking about found treasure and loot, but the cash an adventurer carries on hand. For a bog standard, D&D-style fantasy game, keeping track of money seems like the right thing to do. It fits the model of adventurers who need to scrape together the coin necessary to buy a new sword, spell components, or bundles of crossbow bolts. It also fits the sword-and-sorcery genre, which I'll discuss later.
The strange thing about keeping track of coinage on hand is the problem of large amounts. Thousands of coins can be converted to gems and other high-priced baubles, but then you might run into the issue of paying for smaller daily expenses. Not many taverns would be able (or willing) to make change for that opal you just found in the lich's tomb.
Call of Cthulhu was the first RPG with an abstract money system that I remember. Although equipment and weapons are listed with prices in dollars, money really wasn't an issue -- most likely because treasure hunting wasn't an objective of the game. Unless the characters were going on an expedition or they needed an outlay of cash at that level for some reason, it isn't necessary to keep track of how many dollar bills are in your pocket. At least that's the way we always played.
Additionally, characters have a skill called Credit Rating. This is a measure not of cash and credit reserves, but social status and influence, and the ability to raise cash through loans or even panhandling.
From the 4th. edition (1989):
The players' investigators -- the journalists, authors, professors, and so on -- will need money for their investigations. Their cash can come from (1) wages, royalties, and remittances; (2) from previous savings; and (3) grants, gifts, and loans from various non-governmental sources (government is no granter of largesse in the 1920s).
Money is only a problem if the investigators must perform extended investigations. Incidental investigations can be assumed to take place on weekends, or evenings, or days off. (p. 14)
From the 5th. edition (1992):
Income may or may not be important to your play. Take dollar signs with a grain of salt. Many Keepers never mention money or personal property, and rarely do published scenarios raise the subject. Only in campaigns does earning and spending hold much interest or significance; campaigns, however, can be quite detailed. Learn the Keeper's intent. (p. 24)
Savage Worlds has a similar approach to money, at least for starting characters. In the 1st edition rulebook, we find the following at the end of the character creation section:
Next you need to purchase equipment. Some settings may provide your hero with all the gear he needs. In others, you may be assigned a certain amount of money with which to purchase your starting gear. A list of some common gear and weapons can be found in Chapter Two.
Unless your setting book or GM says otherwise, the standard starting amount is $500. (p. 18)
The only other real mention of money is at the beginning of Chapter Two: Gear.
Cost: Equipment prices are relative both to the starting funds of $500 and to their tech level, so a Springfield musket doesn't really cost $250 in 1862. That's just the "worth" of the weapon relative to the tech level and the typical setting it's intended for. (p. 43)
Monday: Three other approaches.