The National Labor Relations Board sided with unions in several cases involving rules for organizing and representing workers, further riling business groups as the board continues to push through decisions by year’s end.

The board’s three Democrats outvoted the group’s sole Republican member in all three of the cases.

In a case known as Specialty Healthcare, the board decided that the union could seek to organize a group that consists only of nursing assistants at a long-term care facility. That was a blow to the employer, which wanted to include other nonprofessional employees in the unit.

Employer groups had been concerned the board would use the health-care industry case to endorse the formation of “mini-bargaining units” in a range of industries, which they said would allow unions to target small groups of pro-union workers.

In their written decision, the board’s majority suggested that while they were setting a new approach at many health-care facilities, they were simply clarifying longstanding policy in other industries. They said that if an employer thinks a union proposal improperly excludes some workers, the onus is on the employer to prove that the excluded workers share an “overwhelming community of interest” with workers in the proposed unit.

The board’s lone Republican, Brian Hayes, dissented, and business groups said they feared the decision would set a precedent for other industries.

“It seems to be that the board is still going down the route of allowing micro-unions, which from a manufacturing and employer standpoint is going to hinder job creation and could disrupt production,” said Joe Trauger, vice president of human resources policy for the National Association of Manufacturers trade group.

A second case, known as Lamons Gasket Co., involved “card check” elections in which employees sign cards to show their interest in joining a union. In its decision, the board said employees opposed to a union would no longer have the right to immediately challenge an employer’s recognition of a card-check vote. Unions prefer the card-check method to secret-ballot elections.

In the third case, known at UGL-Unicco Service Co., the board bolstered the rights of incumbent unions when a company is sold. It said neither the new owner, nor employees nor rival unions can stage an immediate challenge to the union. Instead, they must give a “reasonable period” and “fair chance” for the union to prove its merits in collective bargaining.

Each of the decisions overturned rulings made in previous years when the board was controlled by Republican appointees, underscoring how the board’s interpretation of labor law tends to flip depending on which party is controlling the White House.

Business groups, management lawyers and Republican lawmakers have accused President Barack Obama’s board appointees of going further than what is typical for Democrats. They say the Obama board is overreaching to bolster unions and union-organizing at a time when union membership has dropped to just 6.9% of the private-sector work force.

“The one common theme throughout these cases is that they were decided in favor of keeping the unions in power and blocking the ability of employees to challenge whether or not the unions still enjoy majority status,” U.S. Chamber of Commerce Senior Vice President of Labor Randel Johnson said in an email.

Kimberly Freeman Brown, executive director of the union-advocacy group American Rights at Work, defended the NLRB, specifically in the Lamons Gasket case. She said the ruling was “nothing more than a return to a process for voluntary recognition that for years worked just fine for employers and employees alike.”

The board’s decisions were issued Tuesday but finalized before longtime board member Wilma Liebman’s term expired as chairman Saturday. Mr. Obama, who appoints the members, named Democrat Mark Pearce—already a board member—as the new chairman over the weekend. Mr. Pearce hasn’t disclosed specific plans but has defended board decisions by saying it was carrying out labor law and ensuring workers have an “unencumbered choice” to unionize.

With Ms. Liebman’s departure, the five-seat board now has just three members, the minimum needed to make major new rules and issue case decisions. The board is set shrink to two when Democrat Craig Becker’s term expires Dec. 31, unless Mr. Obama fills one or both of the vacancies before then.

Congressional Republicans and business groups have already signaled they will try to block any nominees, so the board is hurrying to make decisions by year’s end.

Write to Melanie Trottman at melanie.trottman@wsj.com