Punctuation with US measurements
This is, of course, purely a style issue, and I was somewhat surprised at how few style books directly address the OP's question. The best (and only direct) coverage of the question occurs in Words Into Type, third edition (1974), still a major reference book in U.S. publishing, despite its age. Here is the relevant entry in Words Into Type:
Measures, weights, and other units. Units of measure and weight may be abbreviated in technical copy when they are accompanied by a numeral; never abbreviate units when they follow a spelled-out number. Note that all the following abbreviations are identical in both the singular and the plural and that periods are not usually used (except for in.):
[Examples involving multiple-word units of measure:] at wt, atomic weight; bd ft, board foot; bp, boiling point; Btu, British thermal unit; cc, cubic centimeter; cu ft, cubic foot; cps, cycles per second; FM, frequency modulation; ft-lb, foot-pound; ...
Although the Chicago Manual of Style, fifteenth edition (2003), doesn't have an extensive list of abbreviations for units of measure as Words into Type does, it does touch on the subject of abbreviations for units of measure:
9.18 Abbreviations and symbols. If an abbreviation or a symbol is used for the unit of measure, the quantity is always expressed by a numeral. [Cross reference omitted.]
[Examples:] 3 mi 55 mph 7 h 3 g 35 mm film
But later Chicago thoroughly completely muddies the waters by offering these comments in a subsection on "English Measure":
15.71 Periods. In the rare instances in which abbreviations for English units of measure are used in scientific copy, they are usually set without periods; in nonscientific contexts, periods are customary.
Chicago then goes on (in section 15.73) to endorse such forms as "sq. ft." and "cu. yd." though it doesn't face the ugly music of "ft.-lb."
The Associated Press Stylebook (2002), meanwhile, specifies spelling out both dimensions ("Use figures and spell out inches, feet, yards, etc., to indicate depth, height, length and width.") and weights ("Use figures: The baby weighed 9 pounds, 7 ounces."), so I doubt that it would approve of ft-lb regardless of how you punctuated it.
At the U.S. computer magazines where I've worked, we generally spelled out units of measure except when they appeared in tables, where space was in especially short supply; and in those cases we followed Words Into Type in not including periods to punctuate the abbreviations. It makes sense: If your justification for using an abbreviation instead of spelling a word out is that you need to save space, why squander space on periods?
Units of Weight and Measure, International (Metric) and U.S. Customary, National Bureau of Standards Miscellaneous Publication 286, May 1967, page 10, says, "No period is used with symbols for units." The table of units that follows on that same page includes symbols for both SI (metric) and U.S. customary units without trailing periods, making it clear that the rule applies to both types of units of measure.
The responsibility for the setting and maintaining the rules for the use of units of measure within the Federal Government is assigned to the National Institute of Standards and Technology (NIST), which was the National Bureau of Standards (NBS) prior to 1988. The Federal Government recommends these rules to the private sector and most style guides follow the lead that NIST/NBS sets for representing units of measure.
This rule is old news--NBS policy on this subject has been consistent since it was first set 80 years ago, in 1936. Units of Weight and Measure (U.S. Customary and Metric), Definitions and Tables of Equivalents, National Bureau of Standards Miscellaneous Publication M121, January 1936, page 6, says, "The period is omitted after all abbreviations of units, except where the abbreviation forms an English word." As with more recent NBS publications, the table on page 7 in NBS M121 contains both SI (metric) and U.S. customary unit abbreviations without trailing periods (except for "in." for inch), making it clear that the rule applies to both types of units of measure.